We now move onto Step REF . The key to showing that Step REF  holds is to cast the \(\mathcal {R}_n\)  term into an empirical process theory framework and apply the approximation results of [1]}. Let's setup some standard notations. Let \(N(\epsilon , T, \vert \vert \cdot \vert \vert )\)  denote the covering number of radius \(\epsilon > 0\)  for the metric space \((T, \vert \vert \cdot \vert \vert )\) . Let's also denote the uniform entropy integral (see [2]}) for the set \(T\)  equipped with measurable cover \(F\) ,
\(J(\delta , T) := \sup _Q \int _0^\delta \sqrt{ 1 + \log N(\epsilon \vert \vert F\vert \vert _{Q,2}, T, L_2(Q)) } \,\mathrm {d}\epsilon \) 
